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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 14th 2021
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAdded a subsection > ### As a homotopy type > As a [[homotopy type]] the torus is the product of two copies of the [circle](https://ncatlab.org/nlab/show/circle#as_a_homotopy_type). > In [[homotopy type theory]] the torus can be formalized as the [[higher inductive type]] generated by a point `base`, two paths, $p$ and $q$, from `base` to itself, and an element $t$ of $p\cdot q = q \cdot p$. It has been formally shown ([Sojakova15](#Sojakova15)) that this type is equivalent to the product of two circles. For a treatment in [[cubical type theory]], see ([Licata-Brunierie](#Licata-Brunierie)). <a href="https://ncatlab.org/nlab/revision/diff/torus/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/torus/13">v13</a>, <a href="https://ncatlab.org/nlab/show/torus">current</a>

   Added a subsection

   As a homotopy type

   As a homotopy type the torus is the product of two copies of the circle.

   In homotopy type theory the torus can be formalized as the higher inductive type generated by a point base, two paths, $p$ and $q$, from base to itself, and an element $t$ of $p\cdot q = q \cdot p$. It has been formally shown (Sojakova15) that this type is equivalent to the product of two circles. For a treatment in cubical type theory, see (Licata-Brunierie).

   diff, v13, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 4th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded statement of the *stable* homotopy type of the 2-torus, and graphics illustrating this in line with the homotopy types of punctured tori ([here](https://nlab.preschema.com/nlab/show/torus#StableHomotopyType)) <a href="https://ncatlab.org/nlab/revision/diff/torus/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/torus/15">v15</a>, <a href="https://ncatlab.org/nlab/show/torus">current</a>

   added statement of the stable homotopy type of the 2-torus, and graphics illustrating this in line with the homotopy types of punctured tori (here)

   diff, v15, current

3. • CommentRowNumber3.
   • CommentAuthorSamuel Adrian Antz
   • CommentTimeFeb 14th 2024
   • (edited Feb 14th 2024)
   • PermaLink
   Author: Samuel Adrian Antz
   Format: MarkdownItexAdded [[topological complexity]] of [[tori]]. <a href="https://ncatlab.org/nlab/revision/diff/torus/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/torus/21">v21</a>, <a href="https://ncatlab.org/nlab/show/torus">current</a>

   Added topological complexity of tori.

   diff, v21, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeFeb 14th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexIt may be time to create at least a minimum entry on *[[topological complexity]]*, as you keep pointing to it.

   It may be time to create at least a minimum entry on topological complexity, as you keep pointing to it.

5. • CommentRowNumber5.
   • CommentAuthorSamuel Adrian Antz
   • CommentTimeFeb 14th 2024
   • PermaLink
   Author: Samuel Adrian Antz
   Format: MarkdownItexI am already on it. The page will be published within the next hour.

   I am already on it. The page will be published within the next hour.